The unary fragments of metric interval temporal logic: bounded versus lower bound constraints

Authors:
Paritosh K. Pandya;Simoni S. Shah
Affiliations:
Tata Institute of Fundamental Research, Mumbai, India;Tata Institute of Fundamental Research, Mumbai, India
Venue:
ATVA'12 Proceedings of the 10th international conference on Automated Technology for Verification and Analysis
Year:
2012

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Abstract

We study two unary fragments of the well-known metric interval temporal logic $\mathit{MITL[\textsf{U}_I,\textsf{S}_I]}$ that was originally proposed by Alur and Henzinger, and we pin down their expressiveness as well as satisfaction complexities. We show that $\mbox{$\mathit{MITL[\textsf{F}_\infty,\textsf{P}_\infty]}$}$ which has unary modalities with only lower-bound constraints is (surprisingly) expressively complete for Partially Ordered 2-Way Deterministic Timed Automata (po2DTA) and the reduction from logic to automaton gives us its NP-complete satisfiability. We also show that the fragment $\mbox{$\mathit{MITL[\textsf{F}_b,\textsf{P}_b]}$}$ having unary modalities with only bounded intervals has NEXPTIME-complete satisfiability. But strangely, $\mathit{MITL[\textsf{F}_b,\textsf{P}_b]}$ is strictly less expressive than $\mathit{MITL[\textsf{F}_\infty,\textsf{P}_\infty]}$. We provide a comprehensive picture of the decidability and expressiveness of various unary fragments of MITL.